Spectral expansion for finite temperature two-point functions and clustering
Recently, the spectral expansion of finite temperature two-point functions in integrable quantum field theories was constructed using a finite volume regularization technique and the application of multidimensional residues. In the present work, the original calculation is revisited. By clarifying some details in the residue evaluations, we find and correct some inaccuracies of the previous result. The final result for contributions involving no more than two particles in the intermediate states is presented. The result is verified by proving a symmetry property which follows from the general structure of the spectral expansion, and also by numerical comparison to the discrete finite volume spectral sum. A further consistency check is performed by showing that the expansion satisfies the cluster property up to the order of the evaluation.
Correlation functions play a central role in the formulation of many-body quantum systems. Integrable models presents a unique opportunity to study strongly correlated quantum systems in situations where conventional methods break down. Recent experimental advances resulted in renewed interest in integrable models, since it is now possible to realize certain models with the help of optical and magnetic traps [1, 2, 3, 4] or in low-dimensional magnets [5, 6].
In a recent paper  finite temperature (i.e. thermal) two-point correlation functions were constructed using the exact form factors in 1+1 dimensional integrable models. In an integrable quantum field theory, the basic object is the factorized S-matrix [8, 9]. The matrix elements of the local operators (form factors) satisfy a certain set of equations (the form factor bootstrap equations) which follow from general field theoretical arguments supplemented with the special analytic properties of the S-matrix [10, 11, 12, 13]. Solving these equations gives the form factor functions, which can then be used to construct correlation functions by expanding in the basis formed by the infinite volume asymptotic scattering states.
The form factor expansion of zero-temperature correlations in integrable QFT is very well understood. In general, the series has very good convergence properties in massive models and can be evaluated numerically to any desired precision [12, 14]. However, the problem of thermal correlation functions is much more complicated and has been the subject of active research in the last two decades [15, 16, 17, 18, 19, 20, 21, 22, 23]. The form factor construction of the spectral series is plagued with problems due to the presence of disconnected terms in the expansion, which lead to formally divergent expressions. Following Balog it can be shown that the divergent parts cancel with contributions from the partition function . Nevertheless, it is a very non-trivial task to obtain the correct finite answer. Leclair and Mussardo conjectured an answer for the spectral expansion for one-point and two-point functions in terms of form factors dressed by appropriate occupation number factors containing the pseudo-energy function from the thermodynamical Bethe Ansatz . Their proposal for the two-point function was questioned by Saleur ; on the other hand, in the same paper he also gave a proof of the Leclair-Mussardo formula for one-point functions provided the operator considered is the density of some local conserved charge. By comparison to an alternative proposal , it was also shown that the results obtained by naive regularization are ambiguous .
The idea behind our approach is to use a finite volume setting to regularize the divergences. In [25, 26] this was applied to one-point functions giving a confirmation of the Leclair-Mussardo formula to third order; later a derivation to all orders was also obtained . The crucial point is that finite volume is not an ad hoc, but a physical regulator (since physically realizable systems are always of finite size), therefore one is virtually guaranteed to obtain the correct result when taking the infinite volume limit. The natural small parameter for the finite temperature expansion is the Boltzmann-factor where is the mass gap (which is assumed to be nonzero). The result is an integral series, where the th term represents -particle processes over the Fock-vacuum. The contributions with a low number of particles can be interpreted as disconnected terms of matrix elements calculated in a thermal state with a large number of particles [23, 27]. In this sense the approach is similar to the one used in algebraic Bethe Ansatz [28, 29].
Besides correlation functions, the finite volume regularization can also be applied to numerous other problems. The finite volume form factor approach was extended to boundary operators as well , which was used to compute finite temperature one-point functions of boundary operators . Another application of the bulk finite volume form factors is the construction of one-point functions of bulk operators on a finite interval . It was also used  to construct the form factor perturbation expansion in non-integrable field theories (originally proposed by Delfino et al. ) beyond the leading order. It also turned out that this approach can be applied to quenches in field theory [32, 35, 36].
Regarding thermal two-point functions, the finite volume regularization method was first applied by Essler and Konik [37, 38]; however, their methods do not have any obvious extension to higher order. Despite this shortcoming, their results are very useful as shown by their relevance to inelastic neutron scattering experiments . An independent early calculation of the one-particle–one-particle contribution can also be found in .
In  we developed a systematic method to compute the finite temperature form factor expansion to arbitrary orders. It turns out that the machinery of multidimensional residues provides an appropriate formalism to evaluate higher order corrections systematically, and this was demonstrated to all orders which involve only intermediate states with at most two particles. To verify the result, we applied two consistency checks. The first of them was that the correlator should have a finite limit when the volume is taken to infinity, therefore all terms containing positive powers of the volume had to cancel, which was indeed true. The second one took into account that for some contributions there are two independent ways to arrive at the answer, and agreement between them also provides validating evidence. However, for the term , which contains the contributions when both intermediate states involved in the spectral sum contain two particles, the second one is not available and the first is insufficient to check the structure of the result in detail.
Therefore we decided to provide a numerical evidence, especially since the analytic manipulations themselves are rather tedious and complicated, with many possible sources of mistakes. As it turned out, the result for reported in  is unfortunately incorrect. By further investigation, it turned out that some fine details of the residue calculation needed to be carried out more carefully.
Here we report the correct version of the computation and its result, and present the final formula for the finite temperature two-point function including all contributions with at most two-particle intermediate states. To be confident in our results, we perform several checks. First we check that the result for satisfies a particular symmetry property following from the general form of the spectral expansion. Then we apply a detailed numerical verification of our analytic manipulations, and also verify that the final result satisfies the physically required cluster property.
The layout of the paper is as follows. Section 2 introduces the thermal two-point function and the idea of finite volume regularization. In section 3, the methods to evaluate the resulting spectral expansion is presented. We re-derive the results of , including the correct form of the term and present the full formula of the two-point function including all two-particle contributions. The numerical verification of is performed in Section 4, together with a similar verification for in order to establish a benchmark point for numerical accuracy. In Section 5 we prove that the resulting expansion satisfies the cluster property, and in Section 6 the conclusions are presented. There are also three appendices: Appendix A contains the mathematical formulas used for evaluating the residue contributions, while Appendix B contains the end results of the residue evaluations, which are also necessary for the numerical comparison. The proof of the symmetry property of the contribution is given in Appendix C.
2 Finite volume regularization
2.1 The thermal two-point function
A field theory with finite temperature can be defined using a compact Euclidean (Matsubara) time :
We are interested in the two-point function in dimensional field theories:
A naive spectral sum leads to an ill-defined expression due to the presence of disconnected contributions (cf. e.g. the discussion in ). However, one can put the system in a finite spatial volume with periodic boundary conditions
where denotes the trace over the finite-volume states, is the Hamiltonian in volume . This expression can be expanded inserting two complete sets of states
where the matrix elements of local operators are also taken in the finite volume system. To evaluate it, we need an expression for form factors in finite volume.
2.2 The form factor bootstrap
In a dimensional field theory, the energy and the momentum of an on-shell particle is parametrized by the rapidity variable as and . For the sake of simplicity let us suppose that the spectrum of the model consists of a single particle mass . Incoming and outgoing asymptotic states are defined as:
Integrability leads to factorized scattering, which can be summarized by the relation
where denotes the two-particle amplitude; from this any multi-particle scattering process can be obtained by reordering the particles. States are normalized as:
The form factors of a local operator are defined as
With the help of the crossing relations
all form factors can be expressed in terms of the elementary form factors
where denotes the Lorentz spin of the operator . There is also a further equation related to bound states which we do not need in the sequel.
2.3 Form factors in finite volume
where the are momentum quantum numbers, ordered as by convention. The corresponding energy levels are determined by the Bethe-Yang equations
Defining the two-particle phase shift by the relation
The derivative of will be denoted by
due to unitarity, is an odd and is an even function. We can write
where the quantum numbers take integer/half-integer values for odd/even numbers of particles respectively. Eqns. (2.15) must be solved with respect to the particle rapidities , where the energy (relative to the finite volume vacuum state) can be computed as
up to corrections which decay exponentially with . The density of -particle states in rapidity space can be calculated as
The finite volume behavior of local matrix elements can be given as 
where () are the solutions of the Bethe-Yang equations (2.15) corresponding to the state with the specified quantum numbers () at the given volume . The above relation is valid provided there are no disconnected terms i.e. the left and the right states do not contain particles with the same rapidity, i.e. the sets and are disjoint.
It is easy to see that in the presence of nontrivial scattering there are only two cases when exact equality of (at least some of) the rapidities can occur :
The two states are identical, i.e. and
in which case the corresponding diagonal matrix element can be written as a sum over all bipartite divisions of the set of the particles involved (including the trivial ones when is the empty set or the complete set )
is the -particle Bethe-Yang Jacobi determinant (2.17) involving only the -element subset of the particles, and
is the so-called symmetric evaluation of diagonal multi-particle matrix elements.
Both states are parity symmetric states in the spin zero sector, i.e.
Furthermore, both states must contain one (or possibly more, in a theory with more than one species) particle of zero quantum number. Writing and and defining
the formula for the finite-volume matrix element takes the form
2.4 The form factor expansion using finite volume regularization
Using the finite volume description introduced in subsection 2.3 we can write
and and are the total energies and momenta of the multi-particle states and . The task is to calculate the sum in finite volume and then take the limit .
Similarly for the partition function
with denoting the -particle contribution. The inverse of the partition function is expanded as
Putting this together we can rewrite the expansion as
The first few nontrivial terms are given by
In this way we produce a double series expansions in powers of the variables and . Since these variables are independent, each quantity must have a well-defined limit which we denote as
and we obtain that
A similar reordering was also used for the expansion of the one-point function in powers of , and for the boundary one-point function in . It is evident from (2.23) that the with can be obtained from those with after a trivial exchange of with , with and with .
3 The spectral expansion for finite temperature correlators
To evaluate the finite temperature two-point function, it is necessary to evaluate the summation over two sets of intermediate states. For a given this involves an and an particle state. One can start with any of these; to simplify the calculations, it is best to start with the one containing the smallest number or particles, and do the other later. On the other hand, doing the calculation in the reverse order allows one to cross-check the result .
To evaluate the first summation, a systematic method was given in  based on a multidimensional residue method. Once this is done, all the singularities from the form factors are tamed, and the second summation can be performed by a simple transition from the discrete sum to an integral using the density of states. Then, after assembling using the lower coefficients as in (2.26), and taking the limit the final formula for the contribution can be obtained. Another quick validity check of the calculation is provided by the existence of the infinite volume limit.
3.1 Converting sums to contour integrals
For sums over one-particles states with quantum number we can substitute
and are small closed curves surrounding the solution of
in the complex plane.
For two-particle sums over two-particle states with quantum numbers we can use the multidimensional generalization of the residue theorem to write
where is a multi-contour (a direct product of two curves in the variables and ) surrounding the solution of
where due to the definition (2.13) and take half-integer values, and
Since form factors vanish when any two of their arguments coincide, we can extend the sum by adding the diagonal:
In the next step, the contours are joined together and opened into straight lines, to a product contour whose components in each variable enclose the real axis. However, this can only be done by including other poles (apart from the ones needed for the state summations) in the interior, which come from singularities of the -dependent denominators and of the form factors. These must be classified and subtracted. This procedure was discussed in some detail in , and for one complex variable it is illustrated in fig. 3.1 (for more complex variable it must be performed in each variables separately). We shall only outline it for the case of the contribution, because of the corrections we make to the previous calculation performed in that paper.
3.2 The contribution revisited
The contribution is given by
with the notation
and where 
The rapidities are quantized by
We perform the -sum first and separate it into a diagonal and an off-diagonal piece:
because the finite volume form factor expressions are different for the two types of contributions. In the second term, the prime indicates that the diagonal contributions are excluded.
3.2.1 The diagonal piece
This calculation is exactly the same as in , so we only highlight the main steps. Starting from
Writing the sum in terms of contour integrals, after opening the contours and performing the large limit the diagonal contribution becomes
3.2.2 The non-diagonal part
In the non-diagonal part, one can use
and the prime denotes the omission of the term. We can substitute
since the form factors vanish when any two of their rapidity arguments are identical.
Now we open the contours to encircle the real axis in and . However, that brings more singularities inside the contour whose contribution must then be subtracted. These can be classified as follows:
Spurious QQ-poles. There are two such terms, which come from including the poles with or . Their contribution vanishes for , hence the term ’spurious’. However, they must be included in the numerical tests, therefore we provide their form in eqn. (B.9). Note that the form factors are not singular in this case, although their limits in such points are direction dependent.
QF-poles. In this case one of the integrations has a pole from a form factor, and the other one from a -term:
FF poles. In this case poles in both integrals come from form factors:
The poles of the form factors can be separated by introducing the regular connected part :